Reading a joint bar graph
If a school canteen shows how many cups of tea were sold each morning and each evening as two coloured bars side by side, you can see at a glance which time of day had more sales.
When two sets of data are shown in the same graph with bars of two different colours side by side, the graph is called a joint bar graph. Each group (a day, a month) has two bars that can be compared.
Parts of the graph: The title tells what the graph is about, the horizontal axis shows the groups, the vertical axis shows the numbers and the scale, and the legend tells which colour stands for what.
In the figure: 1 title, 2 vertical axis and scale 1 cm = 10 cups, 3 horizontal axis (days), 4 legend. The morning sales are 40, 55, 35, 60, 50 and the evening sales are 25, 45, 35, 70, 30 cups.
Rule 1: First read the title, the axes, the scale and the legend — Look at these four parts before reading any bar. If you skip the legend, the two colours get swapped.
- Easy: Monday's blue bar reaches 40. Blue stands for morning, so 40 cups of tea were sold on Monday morning.
- Medium: On Tuesday the blue bar is 55 and the orange bar is 45. So 55 cups were sold in the morning and 45 in the evening.
- Exam type: On Friday the blue bar is 50 and the orange bar is 30. By the legend, 50 cups in the morning and 30 in the evening; if the colours are swapped, the answer is wrong.
Rule 2: Value of a bar = height × scale — Measure how many cm tall the bar is and multiply by the value that 1 cm shows.
- Easy: The scale is 1 cm = 10 cups. Thursday's morning bar is 6 cm tall, so 6 × 10 = 60 cups.
- Medium: If the scale is 1 cm = 5 cups, a bar 7 cm tall shows 7 × 5 = 35 cups.
- Exam type: A graph has 1 cm = 20 bicycles and a bar 3.5 cm tall. The value is 3.5 × 20 = 70 bicycles.
Reading a joint bar graph
In the app you can move this figure with your finger.
Open in the appIn the figure: The first bar is between 30 and 40, so 30 + 5 = 35. The second bar is 3 small marks above 40 and each mark is worth 2, so 40 + 3 × 2 = 46.
Rule 3: Reading a value between marks — If the top of a bar lies between two big marks, count how many small parts there are between those marks, find the value of one part and then count.
- Easy: If the top of the bar is exactly halfway between 20 and 30, the value is 25.
- Medium: Two big marks are 10 apart and there are 5 parts between them. 1 part = 10 ÷ 5 = 2. If the bar goes 4 parts above 40, the value is 40 + 4 × 2 = 48.
- Exam type: Big marks are 100 apart with 4 parts between them. 1 part = 100 ÷ 4 = 25. If the bar goes 3 parts above 300, the value is 300 + 3 × 25 = 375.
Look at the scale before reading the value of a bar. In one graph the scale is the same for both colours.
Comparing bars of different groups with each other. Compare only the blue and the orange bar of the same group.
If the scale is 1 cm = 10, multiply the height by 10: 4 cm means 40.
Spot the difference
| Part of the graph | What it tells | Example |
|---|---|---|
| Title | What the graph is about | Tea sales |
| Horizontal axis | The groups | Mon, Tue, Wed |
| Vertical axis | The numbers and the scale | 1 cm = 10 cups |
| Legend | Which colour is what | Blue = morning |
Practice questions
- 1In a graph 1 cm = 10 cups. A bar is 5 cm tall. What is its value?
- 2The scale is 1 cm = 5 and the bar is 8 cm tall. What is the value?
- 3On Thursday 60 cups of tea were sold in the morning and 70 in the evening. If the scale is 1 cm = 10 cups, how tall will the two bars be?
- 4The scale is 1 cm = 20 bicycles and a bar is 4.5 cm tall. What is the value?
- 5Two big marks are 10 apart and there are 5 parts between them. A bar goes 3 parts above 30. What is its value?
- 6The value of a bar is 140 and the scale is 1 cm = 20. How many cm tall should the bar be drawn?
- 7The scale is 1 cm = 50. The blue bar is 6 cm tall and the orange bar is 3.4 cm tall. What are their values?
- 8Big marks are 50 apart and there are 5 parts between them. A bar goes 4 parts above 150. What is its value?
Show answers (try them yourself first)
- 15 × 10 = 50 cups. 2) 8 × 5 = 40. 3) 60 ÷ 10 = 6 cm and 70 ÷ 10 = 7 cm.
- 44.5 × 20 = 90 bicycles. 5) 1 part = 10 ÷ 5 = 2; value 30 + 3 × 2 = 36. 6) 140 ÷ 20 = 7 cm.
- 7Blue 6 × 50 = 300 and orange 3.4 × 50 = 170.
- 81 part = 50 ÷ 5 = 10; value 150 + 4 × 10 = 190.
Comparing two bars
If the sales of two shops in the same month are shown as two bars side by side, you can see at once which shop is ahead and by how much.
Of the two bars in one group, the taller bar shows the bigger value. The difference between two values = bigger value − smaller value. If both bars are equally tall, the values are equal and the difference is 0.
In the figure: In January the sales of shop A are 40 and of shop B are 55. The bar of B is taller, so the difference is 55 − 40 = 15. In April the sales of A are 30 and of B are 60; the difference is 60 − 30 = 30.
Rule 1: A taller bar means a bigger value — Look at the two bars of one group; the taller bar has the bigger value.
- Easy: In January the bar of B is taller than that of A, so 55 > 40.
- Medium: In April the bar of A is shorter than that of B, so 30 < 60 and the sales of A are lower.
- Exam type: In February the sales of A are 35 and of B are 30. Here the bar of A is taller and 35 > 30.
Rule 2: Difference = bigger value − smaller value — First decide which bar is taller and then subtract the smaller value from the bigger value.
- Easy: In January, 55 − 40 = 15.
- Medium: In the tea graph, on Thursday 60 cups in the morning and 70 in the evening. The evening is more, so the difference is 70 − 60 = 10.
- Exam type: In March the sales of A are 50 and of B are 45. A is more, so the difference is 50 − 45 = 5. If you subtract the bigger from the smaller you get a negative number, but a difference is always positive.
Comparing two bars
In the app you can move this figure with your finger.
Open in the appRule 3: Equal bars and who is ahead — If two bars are equally tall, the difference is 0. If there are many groups, check separately who is ahead in each group.
- Easy: On Wednesday 35 cups in the morning and 35 in the evening. The bars are equally tall, so the difference is 35 − 35 = 0.
- Medium: In the tea graph the morning sales are more on 3 days: Monday, Tuesday and Friday. On Wednesday both are equal and on Thursday the evening is more.
- Exam type: In the four months, B is ahead in January and April, and A is ahead in February and March. So A and B are each ahead in 2 months.
Always compare the two bars of the same group. A taller bar shows a bigger value and bars of equal height show equal values.
Subtracting the bigger value from the smaller one: 30 − 60. A difference is always positive: 60 − 30 = 30.
First spot the taller bar and then do "bigger − smaller". Measuring on the scale from the top of the shorter bar to the top of the taller bar also gives the difference.
Spot the difference
| Situation | The two bars | Difference |
|---|---|---|
| The bar of B is taller | 55 and 40 | 55 − 40 = 15 |
| The bar of A is taller | 35 and 30 | 35 − 30 = 5 |
| Both are equal | 35 and 35 | 35 − 35 = 0 |
| B is much taller | 60 and 30 | 60 − 30 = 30 |
Practice questions
- 1In one group the bar of A is 65 and the bar of B is 40. Which is bigger and what is the difference?
- 2Two bars have values 70 and 70. What is the difference?
- 3On one day the morning sales are 45 and the evening sales are 80. By how much is the evening more?
- 4The scale is 1 cm = 10. Two bars are 7 cm and 4.5 cm tall. Find both values and the difference.
- 5In three months the sales of A are 30, 45, 60 and of B are 50, 40, 60. Find the difference in each month and tell in which month both are equal.
- 6In three classes there are 38, 42, 40 boys and 36, 42, 45 girls. In which class are the boys more, in which are they equal and in which are the girls more? Also give the differences.
- 7In five months the sales of A are 25, 40, 35, 50, 45 and of B are 30, 40, 20, 50, 70. In which month is the difference the greatest and how much is it? Which months are equal?
- 8In a graph 1 cm = 10. The bar of A is 9 cm and the bar of B is 7.5 cm tall. Find both values and the difference.
Show answers (try them yourself first)
- 1A is bigger; 65 − 40 = 25. 2) The difference is 70 − 70 = 0. 3) The evening is more; 80 − 45 = 35.
- 47 × 10 = 70 and 4.5 × 10 = 45; difference 70 − 45 = 25.
- 5First month 50 − 30 = 20, second month 45 − 40 = 5, third month 60 − 60 = 0. In the third month both are equal.
- 6In class 1 the boys are more: 38 − 36 = 2. In class 2 they are equal: 42 − 42 = 0. In class 3 the girls are more: 45 − 40 = 5.
- 7The differences are 5, 0, 15, 0, 25. The greatest difference is in the 5th month: 70 − 45 = 25. The 2nd and the 4th month are equal.
- 8A = 9 × 10 = 90 and B = 7.5 × 10 = 75; difference 90 − 75 = 15.
Totals and differences
To decide which shop sold more in four months, looking at one month is not enough. You have to add up all the bars of that shop.
The sum of all the bars of one colour is the total value of that data. The difference of two totals = bigger total − smaller total. If you look at the difference month by month, you can tell in which month the difference is the greatest.
In the figure: The bars of A are 40, 35, 50, 30 and their sum is 40 + 35 + 50 + 30 = 155. The bars of B are 55, 30, 45, 60 and their sum is 55 + 30 + 45 + 60 = 190. The total difference is 190 − 155 = 35. The differences month by month are 15, 5, 5, 30 and the greatest difference is 30 in April.
Rule 1: Total of one colour — Read all the bars of that colour and add them. Do not mix the two colours.
- Easy: In the tea graph the morning sales of Monday and Tuesday are 40 + 55 = 95 cups.
- Medium: The total morning sales of all five days are 40 + 55 + 35 + 60 + 50 = 240 cups.
- Exam type: The total evening sales of all five days are 25 + 45 + 35 + 70 + 30 = 205 cups.
Rule 2: Difference of two totals — Bigger total − smaller total. Always find the total difference from the two sums.
- Easy: The total sales of shop A are 155 and of B are 190. B is more and the difference is 190 − 155 = 35.
- Medium: The total tea sales are 240 cups in the morning and 205 in the evening. The morning is more and the difference is 240 − 205 = 35 cups.
- Exam type: If you simply add the monthly differences 15, 5, 5, 30 you get 15 + 5 + 5 + 30 = 55. But A was ahead in some months, so the total difference is not 55; it is 190 − 155 = 35.
Totals and differences
In the app you can move this figure with your finger.
Open in the appRule 3: Difference month by month and the greatest difference — Find the difference of each group separately and see which of all the differences is the biggest.
- Easy: In January 55 − 40 = 15 and in February 35 − 30 = 5.
- Medium: In March 50 − 45 = 5 and in April 60 − 30 = 30.
- Exam type: The four differences are 15, 5, 5, 30. The biggest of them, 30, comes in April, so the greatest difference is in April.
To find a total, take all the bars of one colour. To find the total difference, subtract only the two totals.
Taking the sum of the monthly differences as the total difference: 15 + 5 + 5 + 30 = 55. The correct answer is 190 − 155 = 35.
First add up each colour and then subtract only once: 190 − 155 = 35.
Spot the difference
| What to find | How | Example |
|---|---|---|
| Total of one colour | Add all the bars | 40 + 35 + 50 + 30 = 155 |
| Difference of two totals | Bigger total − smaller total | 190 − 155 = 35 |
| Difference of one month | Two bars of one group | 60 − 30 = 30 |
| Greatest difference | The biggest of all differences | 30 (April) |
Practice questions
- 1The bars of A are 20, 30, 40, 10. What is their total?
- 2The bars of B are 25, 15, 35. What is their total?
- 3The total of A is 100 and of B is 75. What is the difference?
- 4In four months the sales of A are 45, 50, 35, 60 and of B are 40, 55, 35, 70. Find the total sales of both and their difference.
- 5On three days 18, 24, 30 cups were sold in the morning and 22, 20, 36 in the evening. Find the difference for each day and the total difference.
- 6In a graph 1 cm = 10. The blue bars are 3, 5, 4 cm and the orange bars are 4.5, 2, 6 cm tall. Find the total of each colour and the difference.
- 7In four months the sales of A are 120, 80, 150, 100 and of B are 90, 110, 130, 140. Find the total sales of both and the difference. In which month is the difference the greatest, and in how many months is A ahead?
- 84 bars of one colour are equally tall and their total value is 240. If the scale is 1 cm = 20, how tall should each bar be drawn?
Show answers (try them yourself first)
- 120 + 30 + 40 + 10 = 100. 2) 25 + 15 + 35 = 75. 3) 100 − 75 = 25.
- 4A = 45 + 50 + 35 + 60 = 190 and B = 40 + 55 + 35 + 70 = 200; difference 200 − 190 = 10.
- 5The daily differences are 22 − 18 = 4, 24 − 20 = 4 and 36 − 30 = 6. The totals are 18 + 24 + 30 = 72 and 22 + 20 + 36 = 78; difference 78 − 72 = 6.
- 6Blue 3 × 10 + 5 × 10 + 4 × 10 = 120 and orange 4.5 × 10 + 2 × 10 + 6 × 10 = 125; difference 125 − 120 = 5.
- 7A = 120 + 80 + 150 + 100 = 450 and B = 90 + 110 + 130 + 140 = 470; difference 470 − 450 = 20. The monthly differences are 30, 30, 20, 40, so the greatest is in the 4th month: 140 − 100 = 40. A is ahead in 2 months (the first and the third).
- 8240 ÷ 4 = 60 and 60 ÷ 20 = 3 cm.
Scale and drawing a graph
The height of a sheet of paper is limited. A bar for 90 cups of tea cannot be 90 cm tall on paper. So we first decide how much value 1 cm will show, and this is called the scale.
The scale tells how much value a bar 1 cm tall shows. Height of a bar = value ÷ scale. Choose a scale so that the biggest value fits within the height of the paper, but the bars should not look tiny either.
Rule 1: Height of a bar = value ÷ scale — Read the value and divide it by the scale. The answer is the height in cm.
- Easy: If the value is 80 and 1 cm = 20, the height is 80 ÷ 20 = 4 cm.
- Medium: With 1 cm = 20, the value 120 gives 120 ÷ 20 = 6 cm and the value 100 gives 100 ÷ 20 = 5 cm.
- Exam type: If the value is 45 and 1 cm = 10, the height is 45 ÷ 10 = 4.5 cm. This bar comes exactly halfway between the 4 cm and 5 cm marks.
In the figure: The paper is 10 cm tall and the biggest value is 90. With 1 cm = 5, 90 ÷ 5 = 18 cm, so the bar goes off the paper. With 1 cm = 10, 90 ÷ 10 = 9 cm, so the bar fits. With 1 cm = 50, 90 ÷ 50 = 1.8 cm, so the bar looks very small.
Rule 2: Choosing a good scale — The biggest value ÷ scale must fit in the height of the paper. Take an easy scale: 5, 10, 20 or 50.
- Easy: If the biggest value is 90 and the paper is 10 cm, then 1 cm = 10 is good: 90 ÷ 10 = 9 cm.
- Medium: The biggest value is 180 and the paper is 10 cm. With 1 cm = 10, 180 ÷ 10 = 18 cm, which does not fit. With 1 cm = 20, 180 ÷ 20 = 9 cm, which fits.
- Exam type: If the biggest value is 35 and we take 1 cm = 50, then 35 ÷ 50 = 0.7 cm, a tiny bar. With 1 cm = 5, 35 ÷ 5 = 7 cm, which looks good.
Scale and drawing a graph
In the app you can move this figure with your finger.
Open in the appIn the figure: 1 axes, 2 equal gaps between groups, 3 scale 1 cm = 20, 4 height 80 ÷ 20 = 4 cm. The bars of P are 80, 120, 100 and the bars of Q are 60, 100, 80.
Rule 3: Steps of drawing a graph — First draw the axes, then decide the scale, then draw two bars side by side in each group, colour them and finally write the title and the legend.
- Easy: For P: 80, 120, 100 and Q: 60, 100, 80 with 1 cm = 20, the bars of P are 4, 6, 5 cm and the bars of Q are 3, 5, 4 cm tall.
- Medium: There are 3 groups, June, July and August, and each group has 2 bars, so 3 × 2 = 6 bars in all have to be drawn.
- Exam type: On the vertical axis draw marks from 0 to 120 at gaps of 20: 0, 20, 40, 60, 80, 100, 120. That is 7 marks.
Rule 4: To make the graph look neat — Use bars of equal width, equal gaps between the groups and the same colour for the same data.
- Easy: Draw all bars 1 cm wide and keep an equal gap of 1 cm between two groups.
- Medium: If the blue bar is first in the first group, keep the blue bar first in every group, and write in the legend what blue stands for.
- Exam type: There are 3 groups with 2 bars each and every bar is 1.5 cm wide. The bars need 6 × 1.5 = 9 cm, and 4 gaps of 1 cm each before, between and after the groups need 4 × 1 = 4 cm. The horizontal axis must be 9 + 4 = 13 cm.
Use one scale for all the bars and write it on the graph.
Taking 1 cm = 5 for 90 and trying to fit an 18 cm bar on a 10 cm paper. First work out 90 ÷ 5 = 18 and compare it with the paper.
Work out the biggest value ÷ the height of the paper and take an easy number bigger than that as the scale: 90 ÷ 10 = 9, so 1 cm = 10.
Spot the difference
| Scale | Bar for the value 90 | Result |
|---|---|---|
| 1 cm = 5 | 90 ÷ 5 = 18 cm | Does not fit on the paper |
| 1 cm = 10 | 90 ÷ 10 = 9 cm | Good |
| 1 cm = 20 | 90 ÷ 20 = 4.5 cm | Fits, but a little small |
| 1 cm = 50 | 90 ÷ 50 = 1.8 cm | Too small |
Practice questions
- 1The scale is 1 cm = 10. If the value is 70, how tall should the bar be?
- 2The scale is 1 cm = 5. If the value is 40, what is the height of the bar?
- 3The scale is 1 cm = 20 and the bar is 6 cm tall. What is its value?
- 4The values are 30, 60, 45 and the scale is 1 cm = 15. How tall should the three bars be drawn?
- 5The biggest value is 160 and the paper is 10 cm tall. Is 1 cm = 10 suitable? If not, suggest a suitable scale.
- 6For P: 60, 100, 40 and Q: 80, 20, 100 the scale is 1 cm = 20. Find the height of each bar.
- 7The biggest value is 450 and the paper is 9 cm tall. With which of the scales 10, 20, 50 and 100 does the bar fit on the paper? Which is the best?
- 8A graph has 4 groups with 2 bars each. Each bar is 1 cm wide, and there are 5 gaps of 0.5 cm each before, between and after the groups. How long must the horizontal axis be?
Show answers (try them yourself first)
- 170 ÷ 10 = 7 cm. 2) 40 ÷ 5 = 8 cm. 3) 6 × 20 = 120.
- 430 ÷ 15 = 2 cm, 60 ÷ 15 = 4 cm and 45 ÷ 15 = 3 cm.
- 5No; 160 ÷ 10 = 16 cm, which goes off the paper. With 1 cm = 20, 160 ÷ 20 = 8 cm, which fits.
- 6P: 3, 5, 2 cm and Q: 4, 1, 5 cm (60 ÷ 20 = 3, 100 ÷ 20 = 5, 40 ÷ 20 = 2, 80 ÷ 20 = 4, 20 ÷ 20 = 1).
- 7450 ÷ 10 = 45, 450 ÷ 20 = 22.5, 450 ÷ 50 = 9 and 450 ÷ 100 = 4.5 cm. With 50 and 100 the bar fits; the best is 1 cm = 50, because the bar is 9 cm.
- 8The bars need 4 × 2 × 1 = 8 cm and the gaps need 5 × 0.5 = 2.5 cm; in all 8 + 2.5 = 10.5 cm.
Using graphs in daily life
Joint bar graphs appear in newspapers, on school notice boards and in reports. They make it easy to answer "which is more?", "by how much?" and "when did it increase?".
Drawing conclusions from a graph means telling "which is more, by how much, and when there was an increase or a decrease" from the comparison, the sum and the difference of the bars. Always give conclusions from the numbers in the graph.
Data for the examples: In three weeks a library lent 45, 60, 50 story books and 30, 40, 55 information books to readers.
Rule 1: Which is more and by how much — Look at the two bars of one week, decide which is taller and find the difference. To compare all the weeks together, add them up.
- Easy: In week 1 story books 45 and information books 30. Story books are more and the difference is 45 − 30 = 15.
- Medium: In week 3 information books 55 and story books 50. Here 55 > 50, the difference is 55 − 50 = 5; only in week 3 are information books ahead.
- Exam type: The total of story books is 45 + 60 + 50 = 155 and of information books is 30 + 40 + 55 = 125. 155 − 125 = 30 more story books were lent.
Rule 2: Increase and decrease — Look at the consecutive bars of one colour. Increase or decrease = the difference between the later value and the earlier value.
- Easy: Story books were 45 in week 1 and 60 in week 2. The increase is 60 − 45 = 15.
- Medium: Story books were 60 in week 2 and 50 in week 3. The decrease is 60 − 50 = 10.
- Exam type: Information books rise as 30, 40, 55. The first rise is 40 − 30 = 10 and the second rise is 55 − 40 = 15. The total rise over three weeks is 10 + 15 = 25, that is 55 − 30 = 25.
In the figure: A line graph shows change over time. A pie chart shows the parts of a whole. A histogram shows counts in classes.
Rule 3: Choosing the right graph — Use a joint bar graph to compare two sets of data side by side, a line graph for change over time and a pie chart for the parts of a whole.
- Easy: To compare how many boys and how many girls two classes have, side by side, use a joint bar graph.
- Medium: To show how the temperature changes every hour of the day, a line graph is suitable.
- Exam type: Out of 200 rupees, 100 rupees were spent on food, 60 rupees on travel and 40 rupees on other things. These are parts of a whole, so a pie chart is good; the sum of the parts is 100 + 60 + 40 = 200.
For "which is more?" look at the two bars of one group, and for "when did it increase?" look at the consecutive bars of one colour. Give the numbers and the difference with the answer.
Mixing up the comparison of two colours in one group (difference) with the comparison of one colour with the next group (increase or decrease).
Read the question and decide: for a comparison look inside one group, for a change look ahead along one colour, for a total add one colour.
Spot the difference
| Question | Where to look | Example |
|---|---|---|
| Which is more? | Two bars of one group | 45 and 30 |
| Increase or decrease? | Consecutive bars of one colour | 45 to 60: increase 15 |
| How much in all? | All bars of one colour | 45 + 60 + 50 = 155 |
| Change over time? | Line graph | Temperature during the day |
Practice questions
- 1In week 2 the library lent 60 story books and 40 information books. Which is more and by how much?
- 2Two consecutive bars of one colour are 35 and 50. Is it an increase or a decrease, and by how much?
- 3Which graph is suitable to show the change of temperature during the day?
- 4In a school, in three months 40, 55, 50 children come by cycle and 60, 45, 50 children come on foot. In each month which is more and what is the difference?
- 5From the data of question 4 find the total number of children coming by cycle and on foot, and their difference.
- 6The children coming by cycle are 40, 55, 50. When was there an increase and when a decrease? By how much?
- 7In five days a stall sold glasses of sherbet: 12, 18, 15, 20, 10 in the morning and 15, 14, 15, 26, 5 in the evening. Find the total sales of both times and the difference. Which day shows the greatest difference, and which day shows equal sales?
- 8Choose a graph for three kinds of data: the temperature each day, the parts of a family's expenses, boys and girls of two classes.
Show answers (try them yourself first)
- 1Story books are more; 60 − 40 = 20. 2) An increase; 50 − 35 = 15. 3) A line graph.
- 4First month: those on foot are more, 60 − 40 = 20. Second month: those by cycle are more, 55 − 45 = 10. Third month: both are equal, 50 − 50 = 0.
- 5By cycle 40 + 55 + 50 = 145 and on foot 60 + 45 + 50 = 155; difference 155 − 145 = 10.
- 6From the first to the second month an increase of 55 − 40 = 15, and from the second to the third a decrease of 55 − 50 = 5.
- 7Morning 12 + 18 + 15 + 20 + 10 = 75 and evening 15 + 14 + 15 + 26 + 5 = 75; total difference 75 − 75 = 0. The daily differences are 3, 4, 0, 6, 5, so the greatest is on the 4th day: 26 − 20 = 6. On the 3rd day they are equal: 15 − 15 = 0.
- 8Temperature: a line graph. Parts of the expenses: a pie chart. Boys and girls of two classes: a joint bar graph.